On Average Yield Per dollar

Question

I want to answer the question: On average how many sticks of butter per dollar.

Given the following

        Dollars     Butter     Butter/Dollars
        1           2           2
        1           2           2
        1           2           2
        2           2           1
        2           2           1
        2           3           1.5
        2           3           1.5
        1           3           3

Sum     12          19          14
Avg     1.5         2.375       1.75

AvgDollars/AvgButter = Sum(Dollars)/sum(butter)= 2.375/1.5= 1.58, is my answer I believe.

However, I have no intuition as to what the Avg(Butter/Dollars) = 1.75 represents or why its incorrect. Can anyone explain?

Answer 1

The table just computes the average of the third column, $\frac {14}8=1.75$

It is simply not the case that the average (butter per dollar) is equal to the average(butter)/average(dollar). At least not in general.

Consider the following

$$ \begin{array}{c|lcr} n & \text{Dollars} & \text{Butter} & \text{Butter Per Dollar} \\ \hline 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 1 \end{array} $$

In this instance the two do in fact coincide! Both are $1$.

But now suppose we permute the Butter Column, to get:

$$ \begin{array}{c|lcr} n & \text{Dollars} & \text{Butter} & \text{Butter Per Dollar} \\ \hline 1 & 1 & 2 & 2 \\ 2 & 2 & 1 & \frac 12 \end{array} $$

The ratio of the two averages is still $1$, of course. We haven't changed those averages. But the average(butter per dollar) is now $\frac {2.5}2=1.25$

Answer 2

What you've noticed is the difference $$ \frac{\langle x\rangle}{\langle y\rangle} = \frac{x_1+x_2+\dots +x_n}{y_1+y_2+\dots +y_n} \neq \frac{\frac{x_1}{y_1}+\frac{x_2}{y_2} + \dots \frac{x_n}{y_n}}{n} = \langle\frac x y\rangle $$ where $\langle A\rangle$ denotes the average value of $A$. It shows that the ratio of the averages is generally not equal to the average of the ratio.